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1998 | 127 | 1 | 65-80
Tytuł artykułu

A subsequence characterization of sequences spanning isomorphically polyhedral Banach spaces

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let (x_n) be a sequence in a Banach space X which does not converge in norm, and let E be an isomorphically precisely norming set for X such that (*) ∑_n |x*(x_{n+1} - x_n)| < ∞, ∀x* ∈ E. Then there exists a subsequence of (x_n) which spans an isomorphically polyhedral Banach space. It follows immediately from results of V. Fonf that the converse is also true: If Y is a separable isomorphically polyhedral Banach space then there exists a normalized M-basis (x_n) which spans Y and there exists an isomorphically precisely norming set E for Y such that (*) is satisfied. As an application of this subsequence characterization of sequences spanning isomorphically polyhedral Banach spaces we obtain a strengthening of a result of J. Elton, and an Orlicz-Pettis type result.
Słowa kluczowe
Czasopismo
Rocznik
Tom
127
Numer
1
Strony
65-80
Opis fizyczny
Daty
wydano
1998
otrzymano
1996-10-29
poprawiono
1997-06-09
Twórcy
  • Mathematical Sciences Building, University of Missouri-Columbia, Columbia, Missouri 65211, U.S.A., giorgis@math.missouri.edu
Bibliografia
  • [BP] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164.
  • [D] J. Diestel, Sequences and Series in Banach Spaces, Grad. Texts in Math. 92, Springer, 1984.
  • [DFH] R. Deville, V. Fonf and P. Hájek, Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces, preprint.
  • [E1] J. Elton, Thesis, Yale University, New Haven, Ct.
  • [E2] J. Elton, Extremely weakly unconditionally convergent series, Israel J. Math. 40 (1981), 255-258.
  • [F1] V. P. Fonf, One property of Lindenstrauss-Phelps spaces, Functional Anal. Appl. 13 (1979), 66-67.
  • [F2] V. P. Fonf, Some properties of polyhedral Banach spaces, Funktsional. Anal. i Prilozhen. 14 (4) (1980), 89-90 (in Russian).
  • [F3] V. P. Fonf, Polyhedral Banach spaces, Mat. Zametki 30 (1981), 627-634 (in Russian).
  • [F4] V. P. Fonf, Weakly extreme properties of Banach spaces, Math. Notes 45 (1989), 488-494.
  • [F5] V. P. Fonf, Three characterizations of polyhedral Banach spaces, Ukrainian Math. J. 42 (1990), 1145-1148.
  • [HOR] R. Haydon, E. Odell and H. Rosenthal, On certain classes of Baire-1 functions with applications to Banach space theory, in: Lecture Notes in Math. 1470, Springer, Berlin, 1991, 1-35.
  • [KF] M. Kadets and V. Fonf, Some properties of the set of extreme points of the unit ball of a Banach space, Math. Notes 20 (1976), 737-739.
  • [KP] M. I. Kadets and A. Pełczyński, Basic sequences, biorthogonal systems and norming sets in Banach and Fréchet spaces, Studia Math. 25 (1965), 297-323.
  • [K] V. L. Klee, Polyhedral sections of convex bodies, Acta Math. 4 (1966), 235-242.
  • [L] D. Leung, Some isomorphically polyhedral Orlicz sequence spaces, Israel J. Math. 87 (1994), 117-128.
  • [M] V. D. Milman, Geometric theory of Banach spaces, I, The theory of bases and minimal systems, Russian Math. Surveys 25 (3) (1970), 111-170.
  • [R] H. Rosenthal, Some aspects of the subspace structure of infinite-dimensional Banach spaces, in: Approximation Theory and Functional Analysis, C. Chuy (ed.), Academic Press, New York, 1991, 151-176.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv127i1p65bwm
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